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Theorem imageval 25027
Description: The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
imageval  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
Distinct variable group:    x, R

Proof of Theorem imageval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funimage 25025 . . 3  |-  Fun Image R
2 funrel 5354 . . 3  |-  ( Fun Image R  ->  Rel Image R )
31, 2ax-mp 8 . 2  |-  Rel Image R
4 mptrel 24682 . 2  |-  Rel  (
x  e.  _V  |->  ( R " x ) )
5 vex 2867 . . . . 5  |-  y  e. 
_V
6 vex 2867 . . . . 5  |-  z  e. 
_V
75, 6breldm 4965 . . . 4  |-  ( yImage
R z  ->  y  e.  dom Image R )
8 fnimage 25026 . . . . 5  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
9 fndm 5425 . . . . 5  |-  (Image R  Fn  { x  |  ( R " x )  e.  _V }  ->  dom Image R  =  { x  |  ( R "
x )  e.  _V } )
108, 9ax-mp 8 . . . 4  |-  dom Image R  =  { x  |  ( R " x )  e.  _V }
117, 10syl6eleq 2448 . . 3  |-  ( yImage
R z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
125, 6breldm 4965 . . . 4  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  dom  ( x  e. 
_V  |->  ( R "
x ) ) )
13 eqid 2358 . . . . . 6  |-  ( x  e.  _V  |->  ( R
" x ) )  =  ( x  e. 
_V  |->  ( R "
x ) )
1413dmmpt 5250 . . . . 5  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  e.  _V  |  ( R
" x )  e. 
_V }
15 rabab 2881 . . . . 5  |-  { x  e.  _V  |  ( R
" x )  e. 
_V }  =  {
x  |  ( R
" x )  e. 
_V }
1614, 15eqtri 2378 . . . 4  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  |  ( R "
x )  e.  _V }
1712, 16syl6eleq 2448 . . 3  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
18 imaeq2 5090 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1918eleq1d 2424 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
205, 19elab 2990 . . . 4  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  <->  ( R " y )  e.  _V )
215, 6brimage 25023 . . . . 5  |-  ( yImage
R z  <->  z  =  ( R " y ) )
22 eqcom 2360 . . . . . 6  |-  ( z  =  ( R "
y )  <->  ( R " y )  =  z )
2318, 13fvmptg 5683 . . . . . . . . 9  |-  ( ( y  e.  _V  /\  ( R " y )  e.  _V )  -> 
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  ( R " y ) )
245, 23mpan 651 . . . . . . . 8  |-  ( ( R " y )  e.  _V  ->  (
( x  e.  _V  |->  ( R " x ) ) `  y )  =  ( R "
y ) )
2524eqeq1d 2366 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  z  <-> 
( R " y
)  =  z ) )
26 funmpt 5372 . . . . . . . . 9  |-  Fun  (
x  e.  _V  |->  ( R " x ) )
27 df-fn 5340 . . . . . . . . 9  |-  ( ( x  e.  _V  |->  ( R " x ) )  Fn  { x  |  ( R "
x )  e.  _V } 
<->  ( Fun  ( x  e.  _V  |->  ( R
" x ) )  /\  dom  ( x  e.  _V  |->  ( R
" x ) )  =  { x  |  ( R " x
)  e.  _V }
) )
2826, 16, 27mpbir2an 886 . . . . . . . 8  |-  ( x  e.  _V  |->  ( R
" x ) )  Fn  { x  |  ( R " x
)  e.  _V }
2920biimpri 197 . . . . . . . 8  |-  ( ( R " y )  e.  _V  ->  y  e.  { x  |  ( R " x )  e.  _V } )
30 fnbrfvb 5646 . . . . . . . 8  |-  ( ( ( x  e.  _V  |->  ( R " x ) )  Fn  { x  |  ( R "
x )  e.  _V }  /\  y  e.  {
x  |  ( R
" x )  e. 
_V } )  -> 
( ( ( x  e.  _V  |->  ( R
" x ) ) `
 y )  =  z  <->  y ( x  e.  _V  |->  ( R
" x ) ) z ) )
3128, 29, 30sylancr 644 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  z  <-> 
y ( x  e. 
_V  |->  ( R "
x ) ) z ) )
3225, 31bitr3d 246 . . . . . 6  |-  ( ( R " y )  e.  _V  ->  (
( R " y
)  =  z  <->  y (
x  e.  _V  |->  ( R " x ) ) z ) )
3322, 32syl5bb 248 . . . . 5  |-  ( ( R " y )  e.  _V  ->  (
z  =  ( R
" y )  <->  y (
x  e.  _V  |->  ( R " x ) ) z ) )
3421, 33syl5bb 248 . . . 4  |-  ( ( R " y )  e.  _V  ->  (
yImage R z  <->  y (
x  e.  _V  |->  ( R " x ) ) z ) )
3520, 34sylbi 187 . . 3  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  ->  ( yImage R z  <-> 
y ( x  e. 
_V  |->  ( R "
x ) ) z ) )
3611, 17, 35pm5.21nii 342 . 2  |-  ( yImage
R z  <->  y (
x  e.  _V  |->  ( R " x ) ) z )
373, 4, 36eqbrriv 4864 1  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1642    e. wcel 1710   {cab 2344   {crab 2623   _Vcvv 2864   class class class wbr 4104    e. cmpt 4158   dom cdm 4771   "cima 4774   Rel wrel 4776   Fun wfun 5331    Fn wfn 5332   ` cfv 5337  Imagecimage 24941
This theorem is referenced by:  fvimage  25028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-eprel 4387  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fo 5343  df-fv 5345  df-1st 6209  df-2nd 6210  df-symdif 24920  df-txp 24953  df-image 24963
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